Periodic Trends: Effective Nuclear Charge, Atomic/Ionic Size, Ionization Energy, and Electron Affinity

Trends in the Periodic Table

• This section covers understanding and rationalizing periodic properties in chemistry.
• Key properties discussed include:
  • Effective nuclear charge and shielding
  • Atomic and ionic size
  • Ionization Energy (I.E.)
  • Electron Affinity (E.A.)

Why Periodic Trends?

• Periodic trends are crucial for understanding how atoms interact and form chemical species (e.g., $C$, $H$, $O$) and collections of chemical species (e.g., formaldehyde, $H_2C=O$).
• When atoms connect, several factors are important:
  • How tightly their valence electrons are held: This depends on the effective nuclear charge ($Z_{ ext{eff}}$) and the distance from the nucleus (shell number $n$).
  • Ease of losing an electron: Related to ionization energy.
  • Desire for gaining or sharing an electron: Related to electron affinity.

Shielding & Effective Nuclear Charge

• Definition: Many properties of an atom depend on how tightly its valence electrons are attracted by the nucleus and the amount of positive charge they actually feel.
• Shielding: Electrons situated between an electron of interest and the positively charged nucleus cancel some of the positive nuclear charge.
  • Electrons outside the electron of interest have no effect on its effective nuclear charge.
• Effective Nuclear Charge ($Z_{ ext{eff}}$): The actual positive nuclear charge "felt" by an electron in an atom (other than hydrogen).
  • Formula (approximation): $Z_{ ext{eff}} = Z - S$
    • $Z = ext{atomic number} = ext{nuclear charge}$ (number of protons)
    • $S = ext{number of shielding core electrons}$
  • Usually, core electrons block valence electrons from experiencing the full attraction of the nucleus; this is called "shielding."
  • Example (Lithium, $3_{ ext{Li}}$): $1s^2 2s^1$
    • The $2s$ electron of a $Li$ atom feels a net charge of $+1$ (since $Z=3$ and $S=2$ core electrons, $Z_{ ext{eff}} = 3 - 2 = +1$).
• Key points about $Z_{ ext{eff}}$:
  • Electrons in the same shell do not shield each other effectively.
  • Elements in the same group have similar $Z_{ ext{eff}}$ for their valence electrons.
  • Across a period (left to right): $Z_{ ext{eff}}$ for valence electrons increases because $Z$ increases while the number of core electrons ($S$) remains the same.
  • Down a group (top to bottom): $Z_{ ext{eff}}$ for valence electrons stays almost the same because both $Z$ and $S$ increase proportionally.
  • The distance of an electron from the nucleus (principal quantum number $n$) is also an important factor in the "pull" it feels from the positive nucleus.
  • Distance of the valence electron from the nucleus increases with increasing $n$.
• Example (Iron, Fe, $26_{ ext{Fe}}$): Electron configuration: $1s^2 2s^2 2p^6 3s^2 3p^6 3d^6 4s^2$
  • For a $4s$ electron, $S ext{ (core)} = 2( ext{1s}) + 2( ext{2s}) + 6( ext{2p}) + 2( ext{3s}) + 6( ext{3p}) + 6( ext{3d}) = 24$ (all electrons excluding $4s$).
    • $Z_{ ext{eff}}(4s) = 26 - 24 = +2$ (loosely bound valence electron).
  • For a $3d$ electron, $S ext{ (core)} = 2( ext{1s}) + 2( ext{2s}) + 6( ext{2p}) = 10$ (electrons in shells with n < 3).
    • $Z_{ ext{eff}}(3d) = 26 - 10 = +16$
  • Even though $4s$ electrons have a higher principal quantum number ($n=4$), they are more shielded and feel a smaller $Z_{ ext{eff}}$ compared to $3d$ electrons ($n=3$).

Atomic Size

• The electron cloud around a nucleus has no definite limit, so atomic radius is typically measured as half of the experimentally determined distance between two identical neighboring nuclei in a solid.
  • This measurement applies to metallic elements (half the distance between metal atoms in a metallic crystal) and non-metallic elements (half the distance between nuclei of identical atoms bonded together, often in a molecule).
• Factors affecting atomic size:
  1. Principal quantum number ($n$) for valence electrons: Higher $n$ means larger orbitals.
  2. Effective nuclear charge ($Z<em>{ ext{eff}}$) felt by valence electrons: Higher $Z</em>{ ext{eff}}$ means stronger pull, smaller size.
• Trends in Atomic Radius:
  • Going from top to bottom in a group (e.g., H, Li, Na, K, Rb):
    1. The effective nuclear charge ($Z_{ ext{eff}}$) for the valence electrons stays almost the same.
    2. The principal quantum number ($n$) of the valence electron increases.
    • Result: Atoms become larger because the valence electrons occupy larger orbitals as $n$ increases.
  • Going from left to right across a period (e.g., Na, Mg, Al, Si, P, S, Cl, Ar):
    1. Electrons are added to the same shell ($n$).
    2. The nuclear charge ($Z$) increases, while the number of core electrons ($S$) stays the same; thus, the effective nuclear charge ($Z_{ ext{eff}}$) for the valence electrons increases.
    • Result: Atoms become smaller because the valence electrons are pulled closer to the nucleus as $Z_{ ext{eff}}$ increases.

Ionic Size

• Ions are formed by adding or removing electrons to/from the valence shell.

Cations

• Formation: When an atom loses valence electrons, it forms a positively charged particle called a cation (e.g., $Na$ loses its valence electron to form $Na^{+}$).
• Size comparison to parent atom: Cations are always smaller than their original atoms.
  • Explanation: The number of protons is greater than the number of electrons in the cation, leading to a stronger pull on the remaining electrons, drawing them closer to the nucleus. Also, often the outermost electron shell is lost, reducing the overall size.
• Trends for different cations, same charge:
  • Going from top to bottom in a group: The size of cations with the same charge becomes larger, following the same trend as atomic sizes (due to increasing $n$).
• Trends for different cations, different charges:
  • Going from left to right in a period: The size of cations generally decreases as the nuclear positive charge ($Z$) increases, leading to a stronger pull on the electrons.
• Isoelectronic Cations: Species with the same number of electrons but different numbers of protons.
  • Example: $K^{+} ([<em>{18}Ar]), Ca^{2+} ([</em>{18}Ar]), Sc^{3+} ([_{18}Ar]$ -- all have $18$ electrons).
  • Trend: For isoelectronic cations, the size decreases as the number of protons ($Z$) increases (e.g., K^{+} > Ca^{2+} > Sc^{3+}, because $K$ has $19$ P, $Ca$ has $20$ P, $Sc$ has $21$ P).
  • More protons mean a stronger attraction for the same number of electrons, pulling them in more tightly.
• Same Cation, Different Charges (e.g., Transition Metals):
  • Example (Iron): $Fe$ ($116$ pm) $
    ightarrow$ $Fe^{2+}$ ($76$ pm) $
    ightarrow$ $Fe^{3+}$ ($64$ pm).
  • Removing more electrons from the same atom results in smaller ions.
  • Explanation: With the same number of protons but fewer electrons, the remaining electrons experience a greater effective nuclear charge and are pulled closer to the nucleus.
  • Note for Transition Metals: They always first lose electrons from the outermost shell ($ns$), and then from the inner ($n-1)d$ shell. For example, for $Fe$ ($[_{18}Ar] 4s^2 3d^6$):
    • $Fe^{2+}$: removes $2$ electrons from $4s$ $
      ightarrow$ $[_{18}Ar] 3d^6$
    • $Fe^{3+}$: removes another electron from $3d$ $
      ightarrow$ $[_{18}Ar] 3d^5$

Anions

• Formation: Non-metals tend to gain one or more electrons in their valence shell to achieve the electron configuration of the nearest noble gas, forming a negatively charged particle called an anion (e.g., $Cl$ gains an electron to form $Cl^{-}$).
  • $<em>{17}Cl$ ($[</em>{10}Ne] 3s^2 3p^5$) + $e^{-}$ $
    ightarrow$ $<em>{17}Cl^{-}$ ($[</em>{10}Ne] 3s^2 3p^6$).
• Size comparison to parent atom: Anions are always larger than their parent atoms.
  • Explanation: The number of protons is less than the number of electrons in the anion, so the electrons are less pulled toward the nucleus. Additionally, the increased electron-electron repulsion among the greater number of electrons in the valence shell causes the electron cloud to expand.
• Trends:
  • Going from top to bottom in a group: The size of anions with the same charge becomes larger, similar to atomic and cation trends (due to increasing $n$).

Isoelectronic Species (Anions and Cations)

• Definition: Species (atoms or ions) that have the same number of electrons.
• Trend: The size of isoelectronic species increases as the effective nuclear charge ($Z_{ ext{eff}}$) decreases (meaning fewer protons).
  • Example: For species with $10$ electrons ([Ne] configuration):
    O^{2-} (1.40 ext{ Å}) > F^{-} (1.36 ext{ Å}) > Na^{+} (0.95 ext{ Å}) > Mg^{2+} (0.65 ext{ Å})
  • Explanation: $O^{2-}$ has $8$ protons, $F^{-}$ has $9$ protons, $Na^{+}$ has $11$ protons, $Mg^{2+}$ has $12$ protons. With the same number of electrons, the one with fewer protons ($O^{2-}$) experiences the least nuclear pull and is largest, while the one with the most protons ($Mg^{2+}$) experiences the strongest pull and is smallest.
  • General Order of Size (for isoelectronic series): Anion^{3-} > Anion^{2-} > Anion^{-} > Neutral ext{ }Atom > Cation^{+} > Cation^{2+} > Cation^{3+}
    • Example: N^{3-} > O^{2-} > F^{-} > Na^{+} > Mg^{2+} > Al^{3+}

Ionization Energy (I.E.)

• Definition: The energy (in $kJ/mol$) required to remove one (mole) electron(s) from one (mole) isolated, gaseous atom(s)/ion(s).
  • Equation: X{(g)} ightarrow X^{+}{(g)} + e^{-}
• Meaning: I.E. reflects how tightly an electron is held by the nucleus in an atom.
• Nature: I.E. is always endothermic (requires energy input, thus a positive value). It's also sometimes called "ionization enthalpy."
• First Ionization Energy ($IE_1$): The energy required to remove the most loosely bound electron from a gaseous atom in its ground state.
• Trends in $IE_1$:
  • Going from left to right across a period: $IE<em>1$ increases as the atomic number ($Z$) and effective nuclear charge ($Z</em>{ ext{eff}}$) increase, pulling valence electrons more tightly.
  • Going from top to bottom in a group: $IE_1$ decreases as the principal quantum number ($n$) increases. The valence electrons are further from the nucleus, less attracted, and easier to remove.
  • Relationship with Atomic Size: Trends in I.E. are generally opposite to the trends in atomic size.
    • Larger atoms generally have smaller $IE_1$.
  • Metals vs. Non-metals: Non-metals tend to have the largest $IE<em>1$, while metals tend to have the smallest $IE</em>1$.
• Irregularities in $IE_1$ (e.g., skipping across a period):
  • Example 1: $IE_1$ of Boron (B) is less than Beryllium (Be).
    • $Be: [He] 2s^2$ (filled $2s$ subshell, stable)
    • $B: [He] 2s^2 2p^1$ (removes $2p$ electron)
    • It is easier to remove the $2p$ electron from Boron than a $2s$ electron from Beryllium because the $2p$ electron is in a higher energy subshell and experiences more shielding from the $2s$ electrons. Removing the $2p$ electron from B also results in a stable filled $2s$ subshell configuration for $B^{+}$. This makes $IE<em>1(Be) > IE</em>1(B)$.
  • Example 2: $IE_1$ of Oxygen (O) is less than Nitrogen (N).
    • $N: [He] 2s^2 2p^3$ (half-filled $2p$ subshell, stable)
    • $O: [He] 2s^2 2p^4$ (removes one of the paired $2p$ electrons)
    • It is easier to remove an electron from Oxygen than Nitrogen because in Oxygen, one of the $2p$ orbitals contains a pair of electrons. The electron-electron repulsion between these paired electrons makes it slightly easier to remove one of them, despite Oxygen having a higher nuclear charge. Removing an electron from Oxygen also results in a more stable half-filled $2p^3$ configuration for $O^{+}$. This makes $IE<em>1(N) > IE</em>1(O)$.
• Successive Ionization Energies:
  • Successive ionizations are possible until no electrons remain.
  • Trend: It always requires more energy to remove an electron from an already positive ion (e.g., IE1 < IE2 < IE_3).
    • Example (Lithium):
      • Li{(g)} ightarrow Li^{+}{(g)} + e^{-}: $IE_1 = 513 ext{ kJ/mol}$ (1s^2 2s^1
        ightarrow 1s^2 2s^0)
      • Li^{+}{(g)} ightarrow Li^{2+}{(g)} + e^{-}: $IE_2 = 7298 ext{ kJ/mol}$ (1s^2
        ightarrow 1s^1)
      • Li^{2+}{(g)} ightarrow Li^{3+}{(g)} + e^{-}: $IE_3 = 11815 ext{ kJ/mol}$ (1s^1
        ightarrow 1s^0)
  • Significant Jump: A very large increase in ionization energy occurs when removing a core electron compared to a valence electron. This is because core electrons are much closer to the nucleus and experience a much higher $Z_{ ext{eff}}$. This jump helps identify the number of valence electrons an atom possesses.

Electron Affinity (E.A.)

• Definition: The desire that a gaseous atom/ion has for adding an electron to its valence shell (an "affinity" or "liking" for electrons).
  • Equation: X{(g)} + e^{-} ightarrow X^{-}{(g)}
• Measurement: Electron affinity is measured by the heat or enthalpy change ($riangle EA_H$ in $kJ/mol$) that is absorbed or released when one (mole) electron(s) is added to one (mole) isolated, gaseous atom(s)/ion(s).
• Nature:
  • For almost all elements, adding an electron is an exothermic process ($riangle EA<em>H < 0$), meaning energy is released. A more negative $riangle EA</em>H$ indicates a greater tendency (higher affinity) for an atom to accept an electron.
  • However, for some elements, adding an electron can be an endothermic process ( riangle EA_H > 0), meaning energy is required. This indicates no desire (low affinity) for adding an electron.
• Trends:
  • Going from left to right across a period: Electron affinity generally increases (becomes more negative $riangle EA<em>H$) as the atomic number ($Z$) and effective nuclear charge ($Z</em>{ ext{eff}}$) increase, making the nucleus more attractive to an incoming electron.
  • Non-metals (especially halogens) tend to have high electron affinity (more negative $riangle EA_H$) because gaining an electron allows them to achieve a stable noble gas configuration. Metals generally do not like to form anions.
• Irregularities/Exceptions:
  • Group 2A elements (e.g., Be, Mg), Nitrogen, and Noble Gases have $riangle EA_H > 0$ (positive).
    • Group 2A (e.g., $_4Be: 1s^2 2s^2$): These elements have a completely filled $2s$ subshell, which is a very stable configuration. Adding an electron would require placing it into a higher energy $2p$ orbital, disrupting this stability and increasing electron-electron repulsion, thus requiring energy input.
    • Nitrogen (N: $[He] 2s^2 2p^3$): Nitrogen has a stable, half-filled $2p$ subshell. Adding an electron would destabilize this configuration by creating an electron pair in one of the $2p$ orbitals, leading to electron-electron repulsion which requires energy input.
    • Noble Gases (e.g., Ar: $[Ne] 3s^2 3p^6$): These elements have completely filled valence shells. Adding an electron would require placing it into a new, much higher energy principal shell, which is highly unfavorable and requires significant energy input.
• Successive Electron Affinity:
  • Adding a second electron to an atom (forming a $2-$ ion) is almost always an endothermic process ( riangle EA_H > 0) because it involves adding an electron to an already negatively charged ion.
  • Explanation: The incoming electron experiences strong electrostatic repulsion from the existing negative charge of the anion, requiring energy to overcome this repulsion.
  • Example (Oxygen):
    • O{(g)} + e^{-} ightarrow O^{-}{(g)}: $riangle EA_H = -141 ext{ kJ/mol}$ (exothermic)
    • O^{-}{(g)} + e^{-} ightarrow O^{2-}{(g)}: $riangle EA_H = +703 ext{ kJ/mol}$ (endothermic)
    • Net: O{(g)} + 2e^{-} ightarrow O^{2-}{(g)}: $riangle EA_H = +562 ext{ kJ/mol}$

Summary: Periodic Trends